1 1 ’1

proved by observing that the matrix e = 2 v v 1 is an idempotent such that

νu (e) = 1 ’ e.

Hyperbolic quadratic pairs. By mimicking characterization (??) of hyper-

bolic involutions, we may de¬ne hyperbolic quadratic pairs as follows:

(6.12) De¬nition. A quadratic pair (σ, f ) on a central simple algebra A of even

1

degree over a ¬eld F of arbitrary characteristic is called hyperbolic if 2 deg A ∈

ind(A, σ, f ) or, in other words, if A contains a right ideal I such that

1

dimF I = dimF A, σ(I)I = {0}, f I © Sym(A, σ) = {0}.

2

If char F = 2, the map f is determined by σ and ind(A, σ, f ) = ind(A, σ), hence

(σ, f ) is hyperbolic if and only if σ is hyperbolic. If char F = 2, the involution

σ is symplectic and ind(A, σ, f ) ‚ ind(A, σ); therefore, σ is hyperbolic if (σ, f ) is

hyperbolic.

We proceed to show that the quadratic pair associated to a quadratic space is

hyperbolic if and only if the quadratic space is hyperbolic.

Recall that a quadratic space over a ¬eld F is called hyperbolic if it is isometric

to a space H(U ) = (U — • U, qU ) for some vector space U where U — = HomF (U, F )

and

qU (• + u) = •(u)

for • ∈ U — and u ∈ U . The corresponding symmetric bilinear space is thus the

hyperbolic space denoted H1 (U ) above.

(6.13) Proposition. Let (V, q) be a nonsingular quadratic space of even dimension

over an arbitrary ¬eld F . The corresponding quadratic pair (σ q , fq ) on EndF (V ) is

hyperbolic if and only if the space (V, q) is hyperbolic.

Proof : By (??), the quadratic pair (σq , fq ) is hyperbolic if and only if V contains

a totally isotropic subspace U of dimension dim U = 1 dim V . This condition is

2

equivalent to (V, q) H(U ): see Scharlau [?, p. 12] (if char F = 2) and [?, p. 340]

(if char F = 2).

Hyperbolic quadratic pairs can also be characterized by the existence of certain

idempotents:

78 II. INVARIANTS OF INVOLUTIONS

(6.14) Proposition. A quadratic pair (σ, f ) on a central simple algebra A of even

degree over a ¬eld F of arbitrary characteristic is hyperbolic if and only if A contains

an idempotent e such that

f (s) = TrdA (es) for all s ∈ Sym(A, σ).

Proof : If f (s) = TrdA (es) for all s ∈ Sym(A, σ), then σ(e) = 1 ’ e, by (??),

hence the ideal eA is isotropic for σ and has reduced dimension 1 deg A if e is

2

an idempotent. Moreover, for s ∈ eA © Sym(A, σ) there exists x ∈ A such that

s = ex = σ(x)σ(e), hence

f (s) = TrdA eσ(x)σ(e) = TrdA σ(e)eσ(x) = 0.

Therefore the ideal eA is isotropic for (σ, f ), and (σ, f ) is hyperbolic.

Conversely, suppose that I ‚ A is an isotropic right ideal of reduced dimension

1

2 deg A. By arguing as in (??), we get an idempotent e0 ∈ A such that σ(e0 ) = 1’e0

1

and I = e0 A. If char F = 2, we have e0 ’ 2 ∈ Skew(A, σ) = Alt(A, σ), hence,

1

by (??), TrdA (e0 s) = 2 TrdA (s) = f (s) for all s ∈ Sym(A, σ). We may thus set

e = e0 if char F = 2, and assume that char F = 2 for the rest of the proof.

For all x ∈ Skew(A, σ), the element e = e0 ’ e0 xσ(e0 ) also is an idempotent

such that σ(e) = 1 ’ e. To complete the proof in the case where char F = 2, we

show that e satis¬es the required condition for a suitable choice of x. Consider the

linear form • on Sym(A, σ) de¬ned by

•(s) = f (s) ’ TrdA (e0 s) for s ∈ Sym(A, σ).

This form vanishes on Symd(A, σ), since σ(e0 ) = 1 ’ e0 , and also on I © Sym(A, σ)

since for all s ∈ I © Sym(A, σ) we have s2 = 0, hence TrdA (s) = 0. On the other

hand, for all x ∈ e0 Skew(A, σ)σ(e0 ), the linear form ψx ∈ Sym(A, σ)— de¬ned

by ψx (s) = TrdA (xs) also vanishes on Symd(A, σ), because x ∈ Skew(A, σ), and

on I © Sym(A, σ) because σ(e0 )e0 = 0. If we show that • = ψx for some x ∈

e0 Skew(A, σ)σ(e0 ), then we may set e = e0 + x.

We may thus complete the proof by dimension count: if deg A = n = 2m, it

can be veri¬ed by extending scalars to a splitting ¬eld of A that

dim Symd(A, σ) + I © Sym(A, σ) = mn,

hence the dimension of the space of linear forms on Sym(A, σ) which vanish on

Symd(A, σ) and I © Sym(A, σ) is m. On the other hand, by (??), the kernel of

the map which carries x ∈ e0 Skew(A, σ)σ(e0 ) to ψx ∈ Sym(A, σ)— is the space

Alt(A, σ) © e0 Skew(A, σ)σ(e0 ), and we may compute its dimension over a splitting

¬eld:

1

dim e0 Skew(A, σ)σ(e0 ) = 2 m(m + 1)

and

dim Alt(A, σ) © e0 Skew(A, σ)σ(e0 ) = 1 m(m ’ 1).

2

Therefore, the space of linear forms on Sym(A, σ) which vanish on the intersection

Symd(A, σ) + I © Sym(A, σ) is { ψx | x ∈ e0 Skew(A, σ)σ(e0 ) }, hence • = ψx for

a suitable element x ∈ e0 Skew(A, σ)σ(e0 ).

§6. THE INDEX 79

6.C. Odd-degree extensions. Using the fact that the torsion in the Witt

group of central simple algebras with involution is 2-primary (Scharlau [?]), we show

in this section that involutions which are not hyperbolic do not become hyperbolic

when tensored with a central simple algebra with involution of odd degree, nor after

an odd-degree scalar extension. This last statement generalizes a weak version of

a theorem of Springer; it is due to Bayer-Fluckiger-Lenstra [?].

Since some of the Witt group arguments do not hold in characteristic 2, we

assume that the characteristic of the base ¬eld F is di¬erent from 2 throughout

this subsection.

(6.15) Proposition. Let (A, σ) be a central simple algebra with involution (of any

kind ) over a ¬eld F and let (B, „ ) be a central simple algebra of odd degree with

involution of the ¬rst kind over F . If (A, σ) —F (B, „ ) is hyperbolic, then (A, σ) is

hyperbolic.

Proof : It follows from (??) that the algebra B is split. Let B = EndF (W ) for

some odd-dimensional F -vector space W and let b be a nonsingular symmetric

bilinear form on W such that „ = σb . Similarly, let (A, σ) = EndD (V ), σh for

some hermitian or skew-hermitian space (V, h) over a central division algebra with

involution (D, θ). We then have

(A, σ) —F (B, „ ) = EndD (V — W ), σh—b ,

and it remains to show that (V, h) is hyperbolic if (V — W, h — b) is hyperbolic.

We mimic the proof of Corollary 2.6.5 in Scharlau [?]. Suppose there exists a

non-hyperbolic hermitian or skew-hermitian space (V, h) over (D, θ) which becomes

hyperbolic when tensored by a nonsingular symmetric bilinear space (W, b) of odd

dimension. Among all such examples, choose one where the dimension of W is

minimal. Let dim W = n (≥ 3). We may assume that b has a diagonalization

1, a2 , . . . , an . Since h — b is hyperbolic, we have in the Witt group W » (D, θ)

where » = +1 if h is hermitian and » = ’1 if h is skew-hermitian,

h — a3 , . . . , an = h — ’1, ’a2 .

Since 1, ’a2 — ’1, ’a2 is hyperbolic, it follows that 1, ’a2 — h — a3 , . . . , an

is hyperbolic. By minimality of n, it follows that 1, ’a2 — h is hyperbolic, hence

h h — a2 .

Similarly, we have h h — ai for all i = 2, . . . , n, hence

n · h = h — 1, a2 , . . . , an .

By hypothesis, this form is hyperbolic; therefore, h has odd order in the Witt group

W » (D, θ), contrary to Scharlau™s result [?, Theorem 5.1].

(6.16) Corollary. Let (A, σ) be a central simple algebra with involution (of any

kind ) over a ¬eld F of characteristic di¬erent from 2 and let L/F be a ¬eld extension

of odd degree. If (AL , σL ) is hyperbolic, then (A, σ) is hyperbolic.

Proof : Embed L ’ EndF (L) by mapping x ∈ L to multiplication by x and let ν

be an involution on EndF (L) leaving the image of L elementwise invariant. (The

existence of such an involution ν follows from (??); explicitly, one may pick any

nonzero F -linear map : L ’ F and take for ν the adjoint involution with respect

to the bilinear form b(x, y) = (xy) on L.) If (AL , σL ) is hyperbolic, then (??) shows

that (A, σ) —F EndF (L), ν is hyperbolic, hence (A, σ) is hyperbolic by (??).

80 II. INVARIANTS OF INVOLUTIONS

(6.17) Corollary. Let (A, σ) be a central simple algebra with involution (of any

kind ) over a ¬eld F of characteristic di¬erent from 2 and let L/F be a ¬eld extension

of odd degree. Let u ∈ Sym(A, σ) © A— be a symmetric unit. If there exists v ∈ AL

such that u = vσL (v), then there exists w ∈ A such that u = wσ(w).

Proof : Consider the involution νu on M2 (A) as in (??). The preceding corollary

shows that M2 (A), νu is hyperbolic if M2 (A)L , (νu )L is hyperbolic. Therefore,

the corollary follows from (??).

This result has an equivalent formulation in terms of hermitian forms, which is

the way it was originally stated by Bayer-Fluckiger and Lenstra [?, Corollary 1.14]:

(6.18) Corollary (Bayer-Fluckiger-Lenstra). Let h, h be nonsingular hermitian

forms on a vector space V over a division F -algebra D, where char F = 2. The

forms h, h are isometric if they are isometric after an odd-degree scalar extension

of F .

Proof : The forms h and h are isometric if and only if h ⊥ ’h is hyperbolic, so

that the assertion follows from (??).

§7. The Discriminant

The notion of discriminant considered in this section concerns involutions of

orthogonal type and quadratic pairs. The idea is to associate to every orthogonal

involution σ over a central simple F -algebra a square class disc σ ∈ F — /F —2 , in

such a way that for the adjoint involution σb with respect to a symmetric bilinear

form b, the discriminant disc σb is the discriminant of the form b. If char F = 2,

we also associate to every quadratic pair (σ, f ) an element disc(σ, f ) ∈ F/„˜(F ),

generalizing the discriminant (Arf invariant) of quadratic forms.

7.A. The discriminant of orthogonal involutions. Let F be a ¬eld of

arbitrary characteristic. Recall that if b is a nonsingular bilinear form on a vector

space V over F , the determinant of b is the square class of the determinant of the

Gram matrix of b with respect to an arbitrary basis (e1 , . . . , en ) of V :

· F —2 ∈ F — /F —2 .

det b = det b(ei , ej ) 1¤i,j¤n

The discriminant of b is the signed determinant:

disc b = (’1)n(n’1)/2 det b ∈ F — /F —2

where n = dim V .

If dim V is odd, then for ± ∈ F — we have disc(±b) = ± disc b. Therefore, the

discriminant is an invariant of the equivalence class of b modulo scalar factors if

and only if the dimension is even. Since involutions correspond to such equivalence

classes, the discriminant of an orthogonal involution is de¬ned only for central

simple algebras of even degree.

The de¬nition of the discriminant of an orthogonal involution is based on the

following crucial result:

(7.1) Proposition. Let (A, σ) be a central simple algebra with orthogonal involu-

tion over F . If deg A is even, then for any a, b ∈ Alt(A, σ) © A— ,

NrdA (a) ≡ NrdA (b) mod F —2 .

§7. THE DISCRIMINANT 81

Proof : Fix some a, b ∈ Alt(A, σ) © A— . The involution σ = Int(a) —¦ σ is symplectic

by (??). The same proposition shows that ab ∈ Alt(A, σ ) if char F = 2 and ab ∈

Sym(A, σ ) if char F = 2; therefore, it follows from (??) that NrdA (ab) ∈ F —2 .

An alternate proof is given in (??) below.

This proposition makes it possible to give the following de¬nition:

(7.2) De¬nition. Let σ be an orthogonal involution on a central simple algebra A

of even degree n = 2m over a ¬eld F . The determinant of σ is the square class of

the reduced norm of any alternating unit:

det σ = NrdA (a) · F —2 ∈ F — /F —2 for a ∈ Alt(A, σ) © A—

and the discriminant of σ is the signed determinant:

disc σ = (’1)m det σ ∈ F — /F —2 .

The following properties follow from the de¬nition:

(7.3) Proposition. Let A be a central simple algebra of even degree over a ¬eld F

of arbitrary characteristic.

(1) Suppose σ is an orthogonal involution on A, and let u ∈ A— . If Int(u) —¦ σ is

an orthogonal involution on A, then disc Int(u) —¦ σ = NrdA (u) · disc σ.

(2) Suppose σ is a symplectic involution on A, and let u ∈ A— . If Int(u) —¦ σ is an

orthogonal involution on A, then disc Int(u) —¦ σ = NrdA (u).

(3) If A = EndF (V ) and σb is the adjoint involution with respect to some nonsin-

gular symmetric bilinear form b on V , then disc σb = disc b.

(4) Suppose σ is an orthogonal involution on A. If (B, „ ) is a central simple F -

algebra with orthogonal involution, then

disc σ if deg B is odd,

disc(σ — „ ) =

1 if deg B is even.

(5) Suppose σ is a symplectic involution on A. If (B, „ ) is a central simple algebra

with symplectic involution and char F = 2, then disc(σ — „ ) = 1. (If char F = 2,

(??) shows that σ — „ is symplectic.)

(6) Suppose σ is an orthogonal involution on A. If σ is hyperbolic, then disc σ =

1. (Since hyperbolic involutions in characteristic 2 are symplectic or unitary, the

hypotheses imply char F = 2.)

Proof : (??) If Int(u) —¦ σ is an orthogonal involution, then σ(u) = u and

Alt A, Int(u) —¦ σ = u · Alt(A, σ)

by (??). The property readily follows.

(??) It su¬ces to show that u ∈ Alt A, Int(u) —¦ σ . This is clear if char F = 2,

since the condition that σ is symplectic and Int(u) —¦ σ is orthogonal implies σ(u) =

’u, by (??). If char F = 2 we have Alt A, Int(u) —¦ σ = u · Alt(A, σ) by (??) and

1 ∈ Alt(A, σ) by (??).

(??) Let n = 2m = dim V and identify A with Mn (F ) by means of a basis e

of V . Let also be ∈ GLn (F ) be the Gram matrix of the bilinear form b with respect

to the chosen basis e. The involution σb is then given by

σb = Int(b’1 ) —¦ t,

e

82 II. INVARIANTS OF INVOLUTIONS

where t is the transpose involution. It is easily seen that disc t = (’1)m (indeed, it

su¬ces to ¬nd an alternating matrix of determinant 1), hence (??) yields:

disc σb = (’1)m det(b’1 ) · F —2 = disc b.

e

(??) If a ∈ Alt(A, σ) © A— , then a — 1 ∈ Alt(A — B, σ — „ ) © (A — B)— . The

property follows from the relation

NrdA—B (a — 1) = NrdA (a)deg B .

(??) Since „ is symplectic, deg B is even, by (??). The same argument as

in (??) applies to yield a — 1 ∈ Alt(A — B, σ — „ ) satisfying

NrdA—B (a — 1) ∈ F —2 .

(??) Let deg A = 2m and let e ∈ A be an idempotent such that e + σ(e) = 1.

We have rdim(eA) = m, hence, over a splitting ¬eld, e may be represented by a

diagonal matrix

e = diag(1, . . . , 1, 0, . . . , 0).

m m

Since σ(e) = 1 ’ e, we have 2e ’ 1 ∈ Alt(A, σ); on the other hand, over a splitting

¬eld,

2e ’ 1 = diag(1, . . . , 1, ’1, . . . , ’1),

m m

hence NrdA (2e ’ 1) = (’1)m and therefore disc σ = 1.

(7.4) Example. Let Q be a quaternion algebra with canonical involution γ. By

Proposition (??), every orthogonal involution on Q has the form σ = Int(s) —¦ γ

for some invertible s ∈ Skew(Q, γ) F . Proposition (??) shows that disc σ =

’ NrdQ (s) · F —2 . Therefore, if two orthogonal involutions σ = Int(s) —¦ γ and

σ = Int(s ) —¦ γ have the same discriminant, then we may assume that s and s have

the same reduced norm, hence also the same reduced characteristic polynomial

since TrdQ (s) = 0 = TrdQ (s ). Therefore,

s = xsx’1 = NrdQ (x)’1 xsγ(x)

for some x ∈ Q— , and it follows that

σ = Int(x) —¦ σ —¦ Int(x)’1 .

This show that orthogonal involutions on a quaternion algebra are classi¬ed up to

conjugation by their discriminant.

Observe also that if σ = Int(s) —¦ γ has trivial discriminant, then s2 ∈ F —2 .

Since s ∈ F , this relation implies that Q splits, hence quaternion division algebras

do not carry any orthogonal involution with trivial discriminant.

The next proposition may be seen as an analogue of the formula for the deter-

minant of an orthogonal sum of two bilinear spaces. Let (A, σ) be a central simple

F -algebra with orthogonal involution and let e1 , e2 ∈ A be symmetric idempotents

such that e1 + e2 = 1. Denote A1 = e1 Ae1 and A2 = e2 Ae2 . These algebras are

central simple and Brauer-equivalent to A (see (??)). They are not subalgebras of

A, however, since their unit elements e1 and e2 are not the unit 1 of A. The involu-

tion σ restricts to involutions σ1 and σ2 on A1 and A2 . If (A, σ) = EndF (V ), σb

for some vector space V and some nonsingular symmetric, nonalternating, bilinear

form b, then e1 and e2 are the orthogonal projections on some subspaces V1 , V2

§7. THE DISCRIMINANT 83

⊥

such that V = V1 • V2 . The algebras A1 , A2 may be identi¬ed with EndF (V1 ) and

EndF (V2 ), and σ1 , σ2 are the adjoint involutions with respect to the restrictions of

b to V1 and V2 . These restrictions clearly are symmetric, but if char F = 2 one of

them may be alternating. Therefore, in the general case, extension of scalars to a

splitting ¬eld of A shows that σ1 and σ2 are both orthogonal if char F = 2, but one

of them may be symplectic if char F = 2.

(7.5) Proposition. With the notation above,

det σ = det σ1 det σ2

where we set det σi = 1 if σi is symplectic.

Proof : Let ai ∈ Alt(Ai , σi ) for i = 1, 2; then a1 + a2 ∈ Alt(A, σ), and scalar

extension to a splitting ¬eld of A shows that

NrdA (a1 + a2 ) = NrdA1 (a1 ) NrdA2 (a2 ).

This completes the proof, since NrdAi (ai ) · F —2 = det σi if σi is orthogonal, and

NrdAi (ai ) ∈ F —2 if σi is symplectic, by (??).

7.B. The discriminant of quadratic pairs. Let (σ, f ) be a quadratic pair

on a central simple F -algebra of even degree. If char F = 2, the involution σ is

orthogonal and the map f is the restriction of 1 TrdA to Sym(A, σ); we then set

2

det(σ, f ) = det σ ∈ F — /F —2 disc(σ, f ) = disc σ ∈ F — /F —2 ;

and

this is consistent with the property that the discriminant of a quadratic form of

even dimension is equal to the discriminant of its polar bilinear form.

For the rest of this subsection, assume char F = 2. Recall that we write

SrdA : A ’ F for the map which associates to every element in A the coe¬cient

of X deg A’2 in its reduced characteristic polynomial (see (??)). Recall also that

„˜(x) = x2 + x for x ∈ F .

(7.6) Proposition. Let ∈ A be such that f (s) = TrdA ( s) for all s ∈ Sym(A, σ)

(see (??)). For all x ∈ A,

SrdA + x + σ(x) = SrdA ( ) + „˜ TrdA (x) .

Proof : It su¬ces to prove this formula after scalar extension to a splitting ¬eld. We

may therefore assume A = Mn (F ). By (??), we may ¬nd an element a ∈ Mn (F )

such that a + at ∈ GLn (F ) and (σ, f ) = (σa , fa ). Letting g = a + at , we then have

f (s) = tr(g ’1 as) for all s ∈ Sym(A, σ). Since (??) shows that the element is

uniquely determined up to the addition of an element in Alt(A, σ), it follows that

= g ’1 a + m + σ(m) for some m ∈ A,

hence

+ x + σ(x) = g ’1 a + (m + x) + σ(m + x).

Since σ = σa = Int(g ’1 ) —¦ t and g t = g, the right side may be rewritten as

g ’1 a + g ’1 y + g ’1 y t , where y = g(m + x).

As proved in (??), we have s2 (g ’1 a + g ’1 y + g ’1 y t ) = s2 (g ’1 a) + „˜ tr(g ’1 y) ,

hence

+ x + σ(x) = s2 (g ’1 a) + „˜ tr(m + x) = s2 (g ’1 a) + „˜ tr(m) + „˜ tr(x) .

s2

84 II. INVARIANTS OF INVOLUTIONS

In particular, by letting x = 0 we obtain s2 ( ) = s2 (g ’1 a) + „˜ tr(m) , hence the

preceding relation yields

s2 + x + „˜(x) = s2 ( ) + „˜ tr(x) .

(7.7) De¬nition. Let (σ, f ) be a quadratic pair on a central simple algebra A

over a ¬eld F of characteristic 2. By (??), there exists an element ∈ A such that

f (s) = TrdA ( s) for all s ∈ Sym(A, σ), and this element is uniquely determined up

to the addition of an element in Alt(A, σ). The preceding proposition shows that

the element SrdA ( ) + „˜(F ) ∈ F/„˜(F ) does not depend on the choice of ; we may

therefore set

det(σ, f ) = SrdA ( ) + „˜(F ) ∈ F/„˜(F )

and, letting deg A = 2m,

m(m’1)

disc(σ, f ) = det(σ, f ) + ∈ F/„˜(F ).

2

The following proposition justi¬es the de¬nitions above:

(7.8) Proposition. Let (V, q) be a nonsingular quadratic space of even dimension

over an arbitrary ¬eld F . For the associated quadratic pair (σ q , fq ) on EndF (V )

de¬ned in (??),

disc(σq , fq ) = disc q.

Proof : If char F = 2, the proposition follows from (??). For the rest of the proof

we therefore assume that char F = 2. Let dim V = n = 2m and consider a basis

(e1 , . . . , en ) of V which is symplectic for the polar form bq , i.e.,

bq (e2i’1 , e2i ) = 1, bq (e2i , e2i+1 ) = 0 and bq (ei , ej ) = 0 if |i ’ j| > 1.

As observed in (??), an element ∈ EndF (V ) such that fq (s) = Trd( s) for all

s ∈ Sym EndF (V ), σq is given by

m

= •q e2i’1 — e2i’1 q(e2i ) + e2i — e2i q(e2i’1 ) + e2i’1 — e2i

i=1

∼

where •q : V — V ’ EndF (V ) is the standard identi¬cation (??) associated to bq .

’

Furthermore we have, by (??), (??),

m m(m’1)

s2 ( ) = q(e2i’1 )q(e2i ) + ,

i=1 2

and therefore

m

disc(σq , fq ) = q(e2i’1 )q(e2i ) = disc q.

i=1

(7.9) Proposition. The discriminant of any hyperbolic quadratic pair is trivial.

Proof : If char F = 2, the proposition follows from (??). We may thus assume

that char F = 2. Let (σ, f ) be a hyperbolic quadratic pair on a central simple

F -algebra A. By (??), there is an idempotent e such that f (s) = TrdA (es) for all

s ∈ Sym(A, σ); thus

m(m’1)

disc(σ, f ) = SrdA (e) + + „˜(F )

2

where m = 1 deg A. Since e is an idempotent such that rdim(eA) = m, we have

2

PrdA,e (X) = (X ’ 1)m , hence SrdA (e) = m , and therefore disc(σ, f ) = 0.

2

§7. THE DISCRIMINANT 85

The discriminant of the tensor product of a quadratic pair with an involution

is calculated in the next proposition. We consider only the case where char F = 2,

since the case of characteristic di¬erent from 2 reduces to the tensor product of

involutions discussed in (??).

(7.10) Proposition. Suppose char F = 2. Let (σ1 , f1 ) be an orthogonal pair on a

central simple F -algebra A1 of degree n1 = 2m1 and let (A2 , σ2 ) be a central simple

F -algebra with involution of the ¬rst kind, of degree n2 . The determinant and the

discriminant of the orthogonal pair (σ1 — σ2 , f1— ) on A1 — A2 de¬ned in (??) are

as follows:

n2

det(σ1 — σ2 , f1— ) = n2 det(σ1 , f1 ) + m1 ;

2

disc(σ1 — σ2 , f1— ) = n2 disc(σ1 , f1 ).

In particular, if σ2 is symplectic, the discriminant of the canonical quadratic pair

(σ1 — σ2 , f— ) on A1 — A2 is trivial since n2 is even.

Proof : Let 1 ∈ A1 be such that f1 (s1 ) = TrdA1 ( 1 s1 ) for all s1 ∈ Sym(A1 , σ1 ).

We claim that the element = 1 — 1 satis¬es f1— (s) = TrdA1 —A2 ( s) for all s ∈

Sym(A1 — A2 , σ1 — σ2 ). By (??), we have

Sym(A1 — A2 , σ1 — σ2 ) = Symd(A1 — A2 , σ1 — σ2 ) + Sym(A1 , σ1 ) — Sym(A2 , σ2 ),

hence it su¬ces to show

(7.11) TrdA1 —A2 x + σ1 — σ2 (x) = TrdA1 —A2 (x) for x ∈ A1 — A2

and

(7.12) TrdA1 —A2 ( s1 — s2 ) = f1 (s1 ) TrdA2 (s2 )

for s1 ∈ Sym(A1 , σ1 ) and s2 ∈ Sym(A2 , σ2 ). Since σ1 — σ2 (x) and xσ1 — σ2 ( ) have

the same reduced trace, we have

TrdA1 —A2 x + σ1 — σ2 (x) = TrdA1 —A2 + σ 1 — σ2 ( ) x .

Now, σ1 — σ2 ( ) = σ1 ( 1 ) — 1, hence it follows by (??) that + σ1 — σ2 ( ) = 1,

proving (??). To prove (??), it su¬ces to observe

TrdA1 —A2 ( s1 — s2 ) = TrdA1 ( 1 s1 ) TrdA2 (s2 ) = f1 (s1 ) TrdA2 (s2 ),

hence the claim is proved.

The determinant of (σ1 —σ2 , f1— ) is thus represented by SrdA1 —A2 ( ) in F/„˜(F ).

Since

n2

= PrdA1 , 1 (X)n2 ,

PrdA1 —A2 , (X) = NrdA1 —A2 (X ’ — 1) = NrdA1 (X ’ 1)

1

we have SrdA1 —A2 ( ) = n2 SrdA1 ( 1 ) + n2 TrdA1 ( 1 ). The proposition follows,

2

since TrdA1 ( 1 ) = f1 (1) and (??) shows that f1 (1) = m1 .

As for orthogonal involutions (see (??)), quadratic pairs on a quaternion algebra

are classi¬ed by their discriminant:

(7.13) Proposition. Let (γ, f1 ) and (γ, f2 ) be quadratic pairs on a quaternion

algebra Q over a ¬eld F of characteristic 2. If disc(γ, f1 ) = disc(γ, f2 ), then there

exists x ∈ Q— such that f2 = f1 —¦ Int(x).

86 II. INVARIANTS OF INVOLUTIONS

Proof : For i = 1, 2, let i ∈ Q be such that fi (s) = TrdQ ( i s) for all s ∈ Sym(Q, γ).

We have TrdQ ( i ) = fi (1) = 1, by (??), and

disc(γ, fi ) = SrdQ ( i ) + „˜(F ) = NrdQ ( i ) + „˜(F ).

Therefore, the hypothesis yields

NrdQ ( 2 ) = NrdQ ( 1 ) + (±2 + ±) = NrdQ ( + ±)

1

for some ± ∈ F . Since 1 is determined up to the addition of an element in

Alt(Q, γ) = F , we may substitute 1 + ± for 1 , and assume NrdQ ( 2 ) = NrdQ ( 1 ).

The elements 1 , 2 then have the same reduced characteristic polynomial, hence

we may ¬nd x ∈ Q— such that 1 = x 2 x’1 . For s ∈ Sym(Q, γ), we then have

TrdQ ( 2 s) = TrdQ (x’1 1 xs) = TrdQ ( 1 xsx’1 ),

hence f2 (s) = f1 (xsx’1 ).

Our ¬nal result is an analogue of the formula for the discriminant of an orthog-

onal sum of quadratic spaces. Let (σ, f ) be a quadratic pair on a central simple

algebra A over a ¬eld F of characteristic 2, and let e1 , e2 ∈ A be symmetric idem-

potents such that e1 + e2 = 1. As in the preceding section, we let A1 = e1 Ae1 ,

A2 = e2 Ae2 and restrict σ to symplectic involutions σ1 and σ2 on A1 and A2 . The

degrees of A1 and A2 are therefore even. We have Sym(Ai , σi ) = Sym(A, σ) © Ai

for i = 1, 2, hence we may also restrict f to Sym(Ai , σi ) and get a quadratic pair

(σi , fi ) on Ai .

(7.14) Proposition. With the notation above,

disc(σ, f ) = disc(σ1 , f1 ) + disc(σ2 , f2 ).

Proof : For i = 1, 2, let i ∈ Ai be such that fi (s) = TrdAi ( i s) for all s ∈

Sym(Ai , σi ). For s ∈ Sym(A, σ), we have

s = e1 se1 + e1 se2 + σ(e1 se2 ) + e2 se2 ,

hence

f (s) = f1 (e1 se1 ) + TrdA (e1 se2 ) + f2 (e2 se2 ).

Since TrdA (e1 se2 ) = TrdA (se2 e1 ) and e2 e1 = 0, the middle term on the right side

vanishes. Therefore,

f (s) = TrdA1 ( 1 e1 se1 ) + TrdA2 ( 2 e2 se2 ) for all s ∈ Sym(A, σ).

Taking into account the fact that ei i ei = for i = 1, 2, we obtain

i

f (s) = TrdA1 ( 1 s) + TrdA2 ( 2 s) = TrdA ( + 2 )s for all s ∈ Sym(A, σ).

1

We may thus compute det(σ, f ):

det(σ, f ) = SrdA ( + 2) + „˜(F ).

1

Scalar extension to a splitting ¬eld of A shows that PrdA, = PrdA1 , 1 PrdA2 , 2 ,

1+ 2

hence

SrdA ( + 2) = SrdA1 ( 1 ) + SrdA2 ( 2 ) + TrdA1 ( 1 ) TrdA2 ( 2 ).

1

1

Since TrdAi ( i ) = fi ( i ) = deg Ai , by (??), the preceding relation yields

2

1

det(σ, f ) = det(σ1 , f1 ) + det(σ2 , f2 ) + deg A1 deg A2 .

4

The formula for disc(σ, f ) is then easily checked, using that deg A = deg A1 +

deg A2 .

§8. THE CLIFFORD ALGEBRA 87

§8. The Cli¬ord Algebra

Since the Cli¬ord algebra of a quadratic form is not invariant when the quad-

ratic form is multiplied by a scalar, it is not possible to de¬ne a corresponding

notion for involutions. However, the even Cli¬ord algebra is indeed an invariant for

quadratic forms up to similarity, and our aim in this section is to generalize its con-

struction to algebras with quadratic pairs. The ¬rst de¬nition of the (generalized,

even) Cli¬ord algebra of an algebra with orthogonal involution of characteristic

di¬erent from 2 was given by Jacobson [?], using Galois descent. Our approach is

based on Tits™ “rational” de¬nition [?] which includes the characteristic 2 case.

Since our main tool will be scalar extension to a splitting ¬eld, we ¬rst discuss

the case of a quadratic space.

8.A. The split case. Let (V, q) be a nonsingular quadratic space over a

¬eld F of arbitrary characteristic. The Cli¬ord algebra C(V, q) is the factor of

the tensor algebra T (V ) by the ideal I(q) generated by all the elements of the form

v — v ’ q(v) · 1 for v ∈ V . The natural gradation of T (V ) (by natural numbers)

induces a gradation by Z/2Z:

T (V ) = T0 (V ) • T1 (V ) = T (V — V ) • V — T (V — V ) .

Since generators of I(q) are in T0 (V ), the Z/2Z gradation of T (V ) induces a gra-

dation of C(V, q):

C(V, q) = C0 (V, q) • C1 (V, q).

We have dimF C(V, q) = 2dim V and dimF C0 (V, q) = 2(dim V )’1 : see Knus [?,

Ch. IV, (1.5.2)].

The even Cli¬ord algebra C0 (V, q) may also be de¬ned directly as a factor

algebra of T0 (V ) = T (V — V ):

(8.1) Lemma. In the tensor algebra T (V — V ), consider the following two-sided

ideals:

(1) I1 (q) is the ideal generated by all the elements of the form

v — v ’ q(v), for v ∈ V .

(2) I2 (q) is the ideal generated by all the elements of the form

u — v — v — w ’ q(v)u — w, for u, v, w ∈ V .

Then

T (V — V )

C0 (V, q) = .

I1 (q) + I2 (q)

Proof : The inclusion map T (V — V ) ’ T (V ) maps I1 (q) and I2 (q) into I(q); it

therefore induces a canonical epimorphism

T (V — V )

’ C0 (V, q).

I1 (q) + I2 (q)

The lemma follows if we show

T (V — V )

dimF ¤ dimF C0 (V, q).

I1 (q) + I2 (q)

This inequality is easily established by using an orthogonal decomposition of V

into subspaces of dimension 1 (if char F = 2) or of dimension 2 (if char F = 2 and

88 II. INVARIANTS OF INVOLUTIONS

dim V is even) or into one subspace of dimension 1 and subspaces of dimension 2

(if char F = 2 and dim V is odd).

Structure of even Cli¬ord algebras. We recall the structure theorem for

even Cli¬ord algebras:

(8.2) Theorem. Let (V, q) be a nonsingular quadratic space over a ¬eld F of ar-

bitrary characteristic.

(1) If dim V is odd : dim V = 2m + 1, then C0 (V, q) is central simple F -algebra of

degree 2m .

(2) If dim V is even: dim V = 2m, the center of C0 (V, q) is an ´tale quadratic F -

e

algebra Z. If Z is a ¬eld, then C0 (V, q) is a central simple Z-algebra of degree 2m’1 ;

if Z F — F , then C0 (V, q) is the direct product of two central simple F -algebras

of degree 2m’1 . Moreover, the center Z can be described as follows:

(a) If char F = 2, Z F [X]/(X 2 ’ δ) where δ ∈ F — is a representative of the

discriminant: disc q = δ · F —2 ∈ F — /F —2 .

(b) If char F = 2, Z F [X]/(X 2 + X + δ) where δ ∈ F is a representative of

the discriminant: disc q = δ + „˜(F ) ∈ F/„˜(F ).

A proof can be found in Knus [?, Ch. IV] or Lam [?, Ch. 5] (for the case where

char F = 2) or Scharlau [?, Ch. 9] (for the cases where char F = 2 or char F = 2

and dim V even).

For future reference, we recall an explicit description of the Cli¬ord algebra

of hyperbolic quadratic spaces, from which a proof of the theorem above can be

derived by scalar extension.

Let U be an arbitrary ¬nite dimensional vector space over F and let H(U ) =

—

(U • U, qU ) be the hyperbolic quadratic space de¬ned by

qU (• + u) = •(u)

for • ∈ U — and u ∈ U , as in §??.

In order to give an explicit description of the Cli¬ord algebra of H(U ), consider

the exterior algebra U . Collecting separately the even and odd exterior powers

of U , we get a Z/2Z-gradation

U= U• U,

0 1

where

2i 2i+1

U= U and U= U.

0 i≥0 1 i≥0

For u ∈ U , let ∈ EndF ( U ) denote (exterior) multiplication on the left by u:

u

u (x1 § · · · § xr ) = u § x 1 § · · · § xr .

For • ∈ U — , let d• ∈ EndF ( U ) be the unique derivation of U extending •

which is explicitly de¬ned by

r

(’1)i+1 x1 § · · · § xi’1 § xi+1 § · · · § xr •(xi ).

d• (x1 § · · · § xr ) =

i=1

It is clear that and d• interchange the subspaces U and U for all u ∈ U ,

u 0 1

• ∈ U —.

§8. THE CLIFFORD ALGEBRA 89

(8.3) Proposition. The map which carries •+u ∈ U — •U to d• + ∈ EndF ( U)

u

induces an isomorphism

∼

˜ : C H(U ) ’ EndF (

’ U ).

The restriction of this isomorphism to the even Cli¬ord algebra is an isomorphism

∼

˜0 : C0 H(U ) ’ EndF (

’ U ) — EndF ( U ).

0 1

r s

U and • ∈ U — ,

Proof : A computation shows that for ξ ∈ U, · ∈

d• (ξ § ·) = d• (ξ) § · + (’1)r ξ § d• (·).

By applying this formula twice in the particular case where r = 1, we obtain

d2 (u § ·) = u § d2 (·);

• •

s

by induction on s we conclude that d2 = 0. Therefore, for · ∈ U , u ∈ U and

•

—

•∈U ,

2

(d• + u) (·) = d• (u § ·) + u § d• (·) = ·•(u).

By the universal property of Cli¬ord algebras, it follows that the map U — • U ’

EndF ( U ) which carries • + u to d• + u induces an algebra homomorphism

˜ : C H(U ) ’ EndF ( U ). The fact that ˜ is an isomorphism is established

by induction on dim U (see Knus [?, Ch. IV, (2.1.1)]). (Alternately, assuming the

structure theorem for Cli¬ord algebras, injectivity of ˜ follows from the fact that

C H(U ) is simple, and surjectivity follows by dimension count).

Let ˜0 be the restriction of ˜ to C0 H(U ) . Since d• + u exchanges 0 U and

—

1 U for all • ∈ U and u ∈ U , the elements in the image of ˜0 preserve 0 U and

1 U . Therefore, ˜0 maps C0 H(U ) into EndF ( 0 U ) — EndF ( 1 U ). This map

is onto by dimension count.

The canonical involution. For every quadratic space (V, q), the identity map

on V extends to an involution on the tensor algebra T (V ) which preserves the ideal

I(q). It therefore induces an involution „ on the Cli¬ord algebra C(V, q). This

involution is called the canonical involution of C(V, q); it is the unique involution

which is the identity on (the image of) V . The involution „ clearly restricts to an

involution on C0 (V, q) which we denote by „0 and call the canonical involution of

C0 (V, q). The type of this canonical involution is determined as follows:

(8.4) Proposition. (1) If dim V ≡ 2 mod 4, then „0 is unitary.

(2) If dim V ≡ 0 mod 4, then „0 is the identity on the center Z of C0 (V, q). It

is orthogonal if dim V ≡ 0 mod 8 and char F = 2, and symplectic if dim V ≡ 4

mod 8 or char F = 2. (In the case where Z F — F , this means that „0 is of

orthogonal or symplectic type on each factor of C0 (V, q).)

(3) If dim V ≡ 1, 7 mod 8, then „0 is orthogonal if char F = 2 and symplectic if

char F = 2.

(4) If dim V ≡ 3, 5 mod 8, then „0 is symplectic.

Proof : Consider ¬rst the case where dim V is even: dim V = 2m. By extending

scalars, we may assume that (V, q) is a hyperbolic quadratic space. Let (V, q) =

H(U ) for some m-dimensional vector space U , hence C(V, q) EndF ( U ) by (??).

Under this isomorphism, the canonical involution „ on C(V, q) corresponds to the

adjoint involution with respect to some bilinear form on U which we now describe.

90 II. INVARIANTS OF INVOLUTIONS

1

Let : U ’ U be the involution such that u = u for all u ∈ U = U

r

and let s : U ’ F be a nonzero linear map which vanishes on U for r < m.

De¬ne a bilinear form b : U — U ’ F by

b(ξ, ·) = s(ξ § ·) for ξ, · ∈ U.

We have b(·, ξ) = s(ξ § ·) for ξ, · ∈ U . Since ζ = (’1)m(m’1)/2 ζ for ζ ∈ m U ,

it follows that b is symmetric if m ≡ 0, 1 mod 4, and it is skew-symmetric if

m ≡ 2, 3 mod 4. If char F = 2, then U is commutative, is the identity on U

and ξ § ξ = 0 for all ξ ∈ U , hence b is alternating. In all cases, the form b is

nonsingular.

For u ∈ U and ξ, · ∈ U we have u = u, hence

b(u § ξ, ·) = s(ξ § u § ·) = b(ξ, u § ·).

Similarly, for • ∈ U — , ξ, · ∈ U , a simple computation (using the fact that d• is

a derivation on U ) shows that

b d• (ξ), · = b ξ, d• (·) .

Therefore, the adjoint involution σb on EndF ( U ) is the identity on all the en-

domorphisms of the form d• + u . It follows that σb corresponds to the canonical

involution „ under the isomorphism ˜ of (??). In view of the type of b, the invo-

lution „ is orthogonal if m ≡ 0, 1 mod 4 and char F = 2, and it is symplectic in

the other cases.

If m is odd, the complementary subspaces 0 U and 1 U are totally isotropic

for b. Therefore, letting e0 ∈ EndF ( U ) (resp. e1 ∈ EndF ( U )) denote the

projection on 0 U (resp. 1 U ) parallel to 1 U (resp. 0 U ), we have σb (e0 ) = e1 .

It follows that σb exchanges EndF ( 0 U ) and EndF ( 1 U ), hence „0 is unitary.

If m is even, b restricts to nonsingular bilinear forms b0 on 0 U and b1 on 1 U ,

and the restriction of σb to EndF ( 0 U ) — EndF ( 1 U ) is σb0 — σb1 . Since b0 and b1

have the same type as b, the proof is complete in the case where dim V is even.

If dim V is odd: dim V = 2m + 1, we may extend scalars to assume (V, q)

decomposes as

(V, q) [’1] ⊥ (V , q )

for some nonsingular quadratic space (V , q ) of dimension 2m which may be as-

sumed hyperbolic. Considering this isometry as an identi¬cation, and letting e ∈ V

denote a basis element of the subspace [’1] such that q(e) = ’1, we get an isomor-

∼

phism C(V , q ) ’ C0 (V, q) by mapping x ∈ V to e · x ∈ C0 (V, q). If char F = 2,

’

the canonical involution „0 on C0 (V, q) corresponds to the canonical involution „

on C(V , q ) under this isomorphism. Therefore, „0 is symplectic. If char F = 2,

the canonical involution „0 corresponds to Int(ζ) —¦ „ where ζ ∈ C(V , q ) is the

product of the elements in an orthogonal basis of V . As observed above, „ is

orthogonal if m ≡ 0, 1 mod 4 and is symplectic if m ≡ 2, 3 mod 4. On the other

hand, „ (ζ) = (’1)m ζ, hence (??) shows that „0 is orthogonal if m ≡ 0, 3 mod 4

and symplectic if m ≡ 1, 2 mod 4.

(8.5) Proposition. The involutions „ and „0 are hyperbolic if the quadratic space

(V, q) is isotropic.

Proof : If (V, q) is isotropic, it contains a hyperbolic plane; we may thus ¬nd in V

vectors x, y such that q(x) = q(y) = 0 and bq (x, y) = 1. Let e = x · y ∈ C0 (V, q) ‚

§8. THE CLIFFORD ALGEBRA 91

C(V, q). The conditions on x and y imply e2 = e and „ (e) = „0 (e) = 1 ’ e, hence

„ and „0 are hyperbolic, by (??).

8.B. De¬nition of the Cli¬ord algebra. Let (σ, f ) be a quadratic pair on

a central simple algebra A over a ¬eld F of arbitrary characteristic. Our goal is

to de¬ne an algebra C(A, σ, f ) in such a way that for every nonsingular quadratic

space (V, q) (of even dimension if char F = 2),

C EndF (V ), σq , fq C0 (V, q)

where (σq , fq ) is the quadratic pair associated to q by (??). The idea behind the

de¬nition below (in (??)) is that EndF (V ) V — V under the standard iden-

ti¬cation •q of (??); since C0 (V, q) is a factor algebra of T (V — V ), we de¬ne

C EndF (V ), σq , fq as a factor algebra of T EndF (V ) .

Let A denote A viewed as an F -vector space. We recall the “sandwich” iso-

morphism

∼

Sand : A — A ’ EndF (A)

’

such that Sand(a — b)(x) = axb for a, b, x ∈ A (see (??)). We use this isomorphism

to de¬ne a map

σ2 : A — A ’ A — A

as follows: for ¬xed u ∈ A — A the map A ’ A de¬ned by x ’ Sand(u) σ(x)

is linear and therefore of the form Sand σ2 (u) for a certain σ2 (u) ∈ A. In other

words, the map σ2 is de¬ned by the condition

Sand σ2 (u) (x) = Sand(u) σ(x) for u ∈ A — A, x ∈ A.

(8.6) Lemma. Let (V, b) be a nonsingular symmetric bilinear space and let σ b be

its adjoint involution on EndF (V ). The map σ2 on EndF (V ) — EndF (V ) satis¬es

σ2 •b (x1 — x2 ) — •b (x3 — x4 ) = •b (x1 — x3 ) — •b (x2 — x4 )

∼

for x1 , x2 , x3 , x4 ∈ V where •b : V — V ’ EndF (V ) is the standard identi¬cation

’

of (??).

Proof : It su¬ces to see that, for x1 , x2 , x3 , x4 , v, w ∈ V ,

Sand •b (x1 — x3 ) — •b (x2 — x4 ) •b (v — w) =

Sand •b (x1 — x2 ) — •b (x3 — x4 ) •b (w — v) .

This follows from a straightforward computation: the left side equals

•b (x1 — x3 ) —¦ •b (v — w) —¦ •b (x2 — x4 ) = •b (x1 — x4 )b(x3 , v)b(w, x2 )

whereas the right side equals

•b (x1 — x2 ) —¦ •b (w — v) —¦ •b (x3 — x4 ) = •b (x1 — x4 )b(x2 , w)b(v, x3 ).

Let ∈ A be such that f (s) = TrdA ( s) for all s ∈ Sym(A, σ). The existence

of such an element is proved in (??), where it is also proved that is uniquely

determined up to the addition of an element in Alt(A, σ). If = + a ’ σ(a) for

some a ∈ A, then for all u ∈ A — A such that σ2 (u) = u we have

Sand(u)( ) = Sand(u)( ) + Sand(u)(a) ’ Sand(u) σ(a) .

92 II. INVARIANTS OF INVOLUTIONS

The last term on the right side is equal to Sand σ2 (u) (a) = Sand(u)(a), hence

Sand(u)( ) = Sand(u)( ).

Therefore, the following de¬nition does not depend on the choice of :

(8.7) De¬nition. The Cli¬ord algebra C(A, σ, f ) is de¬ned as a factor of the ten-

sor algebra T (A):

T (A)

C(A, σ, f ) =

J1 (σ, f ) + J2 (σ, f )

where

(1) J1 (σ, f ) is the ideal generated by all the elements of the form s ’ f (s) · 1, for

s ∈ A such that σ(s) = s;

(2) J2 (σ, f ) is the ideal generated by all the elements of the form u ’ Sand(u)( ),

for u ∈ A — A such that σ2 (u) = u and for ∈ A as above.

The following proposition shows that the de¬nition above ful¬lls our aim:

(8.8) Proposition. Let (V, q) be a nonsingular quadratic space (of even dimen-

sion if char F = 2) and let (σq , fq ) be the associated quadratic pair. The standard

∼

identi¬cation •q : V — V ’ EndF (V ) of (??) induces an identi¬cation

’

∼

·q : C0 (V, q) ’ C EndF (V ), σq , fq .

’

Proof : It su¬ces to show that the isomorphism of tensor algebras

∼

T (•q ) : T (V — V ) ’ T EndF (V )

’

maps I1 (q) to J1 (σq , fq ) and I2 (q) to J2 (σq , fq ).

The ideal T (•q ) I1 (q) is generated by all the elements of the form

•q (v — v) ’ q(v) · 1, for v ∈ V .

Since σq corresponds to the switch map on V — V , the elements s ∈ EndF (V ) such

that σq (s) = s are spanned by elements of the form •q (v — v). Since moreover

q(v) = fq —¦ •q (v — v) by (??), it follows that J1 (σq , fq ) = T (•q ) I1 (q) .

Similarly, (??) shows that the elements u ∈ EndF (V ) — EndF (V ) such that

σ2 (u) = u are spanned by elements of the form •q (x — y) — •q (y — z) for x, y,

z ∈ V . Therefore, in order to show that J2 (σq , fq ) = T (•q ) I2 (q) , it su¬ces to

prove

(8.9) Sand •q (x — y) — •q (y — z) ( ) = q(y)•q (x — z) for x, y, z ∈ V .

1

If char F = 2 we may choose = 2 , hence

Sand •q (x — y) — •q (y — z) ( ) = 1 •q (x — y) —¦ •q (y — z).

2

The right side can be evaluated by (??):

1

— y) —¦ •q (y — z) = 1 bq (y, y)•q (x — z) = q(y)•q (x — z),

2 •q (x 2

proving (??) when char F = 2.

Suppose next char F = 2, hence dim V is even. We may of course assume y = 0

in (??). Let dim V = n = 2m and let (e1 , . . . , en ) be a symplectic basis of V such

that e1 = y. We thus assume

bq (e2i’1 , e2i ) = 1, bq (e2i , e2i+1 ) = 0 and bq (ei , ej ) = 0 if |i ’ j| > 1.

§8. THE CLIFFORD ALGEBRA 93

As observed in (??), we may then choose

m

= •q ( i=1 e2i’1 — e2i’1 q(e2i ) + e2i — e2i q(e2i’1 ) + e2i’1 — e2i ).

By (??) we have

Sand •q (x — e1 ) — •q (e1 — z) •q (e2i’1 — e2i’1 ) =

•q (x — e1 ) —¦ •q (e2i’1 — e2i’1 ) —¦ •q (e1 — z) = 0

for i = 1, . . . , m, and similarly

Sand •q (x — e1 ) — •q (e1 — z) •q (e2i’1 — e2i ) = 0

for i = 1, . . . , m. Moreover,

•q (x — z) for i = 1,

Sand •q (x — e1 ) — •q (e1 — z) •q (e2i — e2i ) =

0 for i > 1,

hence

Sand •q (x — e1 ) — •q (e1 — z) ( ) = q(e1 )•q (x — z).

If char F = 2, the quadratic pair (σ, f ) is entirely determined by the involution

σ, since f (s) = 1 TrdA (s) for all s ∈ Sym(A, σ). We then simply write C(A, σ, f )

2

for C(A, σ) Since we may choose = 1/2, we have Sand(u)( ) = 1 µ(u) where 2

µ : A — A ’ A is the multiplication map: µ(x — y) = xy for x, y ∈ A.

Examples. Cli¬ord algebras of quadratic pairs on nonsplit central simple al-

gebras are not easy to describe explicitly in general. We have the following results

however:

(a) For algebras of degree 2, it readily follows from (??) below that C(A, σ, f )

is the ´tale quadratic F -algebra determined by the discriminant disc(σ, f ).

e

(b) For the tensor product of two quaternion algebras Q1 , Q2 with canonical

involutions γ1 , γ2 it is shown in (??) below that

C(Q1 — Q2 , γ1 — γ2 ) Q1 — Q2 .

More generally, Tao [?] has determined (in characteristic di¬erent from 2)

up to Brauer-equivalence the components of the Cli¬ord algebra of a tensor

product of two central simple algebras with involution: see the notes at the

end of this chapter.

(c) Combining (??) and (??), one sees that the Cli¬ord algebra of a hyper-

bolic quadratic pair on a central simple algebra A of degree divisible by 4

decomposes into a direct product of two central simple algebras, of which

one is split and the other is Brauer-equivalent to A.

Besides the structure theorem in (??) below, additional general information on

Cli¬ord algebras of quadratic pairs is given in (??).

Structure of Cli¬ord algebras. Although the degree of A is arbitrary in

the discussion above (when char F = 2), the case where deg A is odd does not

yield anything beyond the even Cli¬ord algebras of quadratic spaces, since central

simple algebras of odd degree with involutions of the ¬rst kind are split (see (??)).

Therefore, we shall discuss the structure of Cli¬ord algebras only in the case where

deg A = n = 2m.

94 II. INVARIANTS OF INVOLUTIONS

(8.10) Theorem. Let (σ, f ) be a quadratic pair on a central simple algebra A

of even degree n = 2m over a ¬eld F of arbitrary characteristic. The center of

C(A, σ, f ) is an ´tale quadratic F -algebra Z. If Z is a ¬eld, then C(A, σ, f ) is a

e

central simple Z-algebra of degree 2m’1 ; if Z F — F , then C(A, σ, f ) is a direct

product of two central simple F -algebras of degree 2m’1 . Moreover, the center Z is

as follows:

F [X]/(X 2 ’ δ) where δ ∈ F — is a representative of the

(1) If char F = 2, Z

discriminant: disc(σ, f ) = disc σ = δ · F —2 ∈ F — /F —2 .

(2) If char F = 2, Z F [X]/(X 2 + X + δ) where δ ∈ F is a representative of the

discriminant: disc(σ, f ) = δ + „˜(F ) ∈ F/„˜(F ).

Proof : Let L be a splitting ¬eld of A in which F is algebraically closed (for in-

stance the function ¬eld of the Severi-Brauer variety of A). There is a nonsingular

quadratic space (V, q) over L such that

(AL , σL , fL ) EndL (V ), σq , fq ,

by (??). Moreover, (??) shows that disc(σL , fL ) = disc q (in L— /L—2 if char F = 2,

in L/„˜(L) if char F = 2). If δ ∈ F is a representative of disc(σ, f ), we then have

δ · L—2 if char F = 2,

disc q =

δ + „˜(L) if char F = 2.

It is clear from the de¬nition that the construction of the Cli¬ord algebra commutes

with scalar extension, hence by (??)

C(A, σ, f ) —F L = C(AL , σL , fL ) C0 (V, q).

In particular, it follows that the center Z of C(A, σ, f ) is a quadratic ´tale F -

e

algebra which under scalar extension to L becomes isomorphic to L[X]/(X 2 ’ δ) if

char F = 2 and to L[X]/(X 2 + X + δ) if char F = 2. Since F is algebraically closed

in L, it follows that Z F [X]/(X 2 ’ δ) if char F = 2 and Z F [X]/(X 2 + X + δ)

if char F = 2. The other statements also follow from the structure theorem for even

Cli¬ord algebras of quadratic spaces: see (??).

Alternate methods of obtaining the description of Z proven above are given

in (??) and (??).

The canonical involution. Let σ : T (A) ’ T (A) be the involution induced

by σ on the tensor algebra T (A); thus, for a1 , . . . , ar ∈ A,

σ(a1 — · · · — ar ) = σ(ar ) — · · · — σ(a1 ).

Direct computations show that the ideals J1 (σ, f ) and J2 (σ, f ) are preserved un-

der σ. Therefore, σ induces an involution on the factor algebra C(A, σ, f ) which

we also denote by σ and call the canonical involution of C(A, σ, f ).

The following result justi¬es this de¬nition:

(8.11) Proposition. Let (V, q) be a nonsingular quadratic space (of even dimen-

sion if char F = 2) and let (σq , fq ) be the associated quadratic pair on EndF (V ).

Under the standard identi¬cation

∼

·q : C0 (V, q) ’ C EndF (V ), σq , fq

’

of (??), the canonical involution „0 of C0 (V, q) corresponds to the involution σq of

C EndF (V ), σq , fq .

§8. THE CLIFFORD ALGEBRA 95

Proof : The canonical involution „ of C(V, q) is induced by the involution of T (V )

which is the identity on V . Therefore, „0 is induced by the involution of T (V — V )

which switches the factors in V — V . Under the standard identi¬cation of (??), this

involution corresponds to σq .

By extending scalars to a splitting ¬eld of A, we may apply the preceding

proposition and (??) to determine the type of the involution σ on C(A, σ, f ). As

in (??), we only consider the case of even degree.

(8.12) Proposition. Let (σ, f ) be a quadratic pair on a central simple algebra A

of even degree n = 2m over a ¬eld F . The canonical involution σ of C(A, σ, f ) is

unitary if m is odd, orthogonal if m ≡ 0 mod 4 and char F = 2, and symplectic

if m ≡ 2 mod 4 or char F = 2. (In the case where the center of C(A, σ, f ) is

isomorphic to F — F , this means that σ is of orthogonal or symplectic type on each

factor of C(A, σ, f ).)

8.C. Lie algebra structures. We continue with the same notation as in the

preceding section; in particular, (σ, f ) is a quadratic pair on a central simple alge-

bra A over a ¬eld F of arbitrary characteristic and C(A, σ, f ) is the corresponding

Cli¬ord algebra.

Since C(A, σ, f ) is de¬ned as a quotient of the tensor algebra T (A), the canon-

ical map A ’ A ’ T (A) yields a canonical map

(8.13) c : A ’ C(A, σ, f )

which is F -linear but not injective (nor an algebra homomorphism), since c(s) =

1

f (s) for all s ∈ Sym(A, σ). In particular, (??) shows that c(1) = 2 deg A. We will

show that the subspace c(A) ‚ C(A, σ, f ) is a Lie subalgebra of L C(A, σ, f ) , and

relate it to the Lie subalgebra Alt(A, σ) ‚ L(A).

(8.14) Lemma. The kernel of c is ker c = ker f ‚ Sym(A, σ), and dim c(A) =

n(n’1)

+ 1 if deg A = n. Moreover, for x1 , x2 ∈ A we have

2

c(x1 ), c(x2 ) = c x1 ’ σ(x1 ), x2

where [ , ] are the Lie brackets.

Proof : Since c and f have the same restriction to Sym(A, σ), it is clear that

ker f ‚ ker c. Dimension count shows that this inclusion is an equality if we show

dim c(A) = n(n’1) + 1.

2

In order to compute the dimension of c(A), we may extend scalars to a splitting

¬eld of A. Therefore, it su¬ces to consider the case where A is split: let A =

EndF (V ) and (σ, f ) = (σq , fq ) for some nonsingular quadratic space (V, q) (of even

∼

dimension if char F = 2). Under the standard identi¬cations •q : V — V ’ A ’

∼

of (??) and ·q : C0 (V, q) ’ C(A, σ, f ) of (??), the map c : V — V ’ C0 (V, q) is

’

given by the multiplication in C(V, q):

c(v — w) = v · w ∈ C0 (V, q) for v, w ∈ V .

Let (e1 , . . . , en ) be an arbitrary basis of V . The Poincar´-Birkho¬-Witt theorem

e

(Knus [?, Ch. IV, (1.5.1)]) shows that the elements 1 and ei · ej for i < j are

linearly independent in C0 (V, q). Since these elements span c(V — V ), it follows

that dim c(V — V ) = n(n’1) + 1, completing the proof of the ¬rst part.

2

In order to prove the last relation, we may also assume that A is split. As

above, we identify A with V — V by means of •q . Since both sides of the relation

96 II. INVARIANTS OF INVOLUTIONS

are bilinear in x1 , x2 , it su¬ces to prove it for x1 = v1 — w1 and x2 = v2 — w2 with

v1 , v2 , w1 , w2 ∈ V . Then

x1 ’ σ(x1 ), x2 = (v1 — w1 ’ w1 — v1 ) —¦ (v2 — w2 )

’ (v2 — w2 ) —¦ (v1 — w1 ’ w1 — v1 )

= v1 — w2 bq (w1 , v2 ) ’ w1 — w2 bq (v1 , v2 )

’ v2 — w1 bq (w2 , v1 ) + v2 — v1 bq (w2 , w1 ),

hence

c [x1 ’ σ(x1 ), x2 ] = v1 · w2 bq (w1 , v2 ) ’ w1 · w2 bq (v1 , v2 )

+ v2 · v1 bq (w2 , w1 ) ’ v2 · w1 bq (w2 , v1 ).

For u, v ∈ V , we have u · v + v · u = bq (u, v); therefore, the four terms on the right

side of the last equation can be evaluated as follows:

v1 · w2 bq (w1 , v2 ) = v1 · w1 · v2 · w2 + v1 · v2 · w1 · w2

w1 · w2 bq (v1 , v2 ) = v1 · v2 · w1 · w2 + v2 · v1 · w1 · w2

v2 · v1 bq (w2 , w1 ) = v2 · v1 · w1 · w2 + v2 · v1 · w2 · w1

v2 · w1 bq (w2 , v1 ) = v2 · v1 · w2 · w1 + v2 · w2 · v1 · w1 .

The alternating sum of the right sides is

v1 · w1 · v2 · w2 ’ v2 · w2 · v1 · w1 = c(x1 ), c(x2 ) .

The lemma shows that c(A) is stable under the Lie brackets, and is therefore a

Lie subalgebra of L C(A, σ, f ) . Moreover, it shows that if x, y ∈ A are such that

c(x) = c(y), then x ’ y ∈ Sym(A, σ), hence x ’ σ(x) = y ’ σ(y). We may therefore

de¬ne a map

δ : c(A) ’ Alt(A, σ)

by

δ c(x) = x ’ σ(x) for x ∈ A.

(8.15) Proposition. The map δ is a Lie-algebra homomorphism which ¬ts into

an exact sequence

δ

0 ’ F ’ c(A) ’ Alt(A, σ) ’ 0.

’

Proof : For x, y ∈ A we have c(x), c(y) = c x ’ σ(x), y by (??), hence

δ c(x), c(y) = x ’ σ(x), y ’ σ x ’ σ(x), y = x ’ σ(x), y ’ σ(y) ,

proving that δ is a Lie-algebra homomorphism. This map is surjective by de¬nition.

In order to show F ‚ ker δ, pick an element a ∈ A such that TrdA (a) = 1; we then

have

c a + σ(a) = f a + σ(a) = TrdA (a) = 1,

hence

δ(1) = a + σ(a) ’ σ a + σ(a) = 0.

Therefore, F ‚ ker δ, and dimension count shows that this inclusion is an equality.

§8. THE CLIFFORD ALGEBRA 97

We proceed to de¬ne on c(A) another Lie-algebra homomorphism, using the

canonical involution σ on C(A, σ, f ).

(8.16) Lemma. For all x ∈ A,

σ c(x) = c σ(x) c(x) + σ c(x) = TrdA (x).

and

In particular, Id + σ maps c(A) onto F . Therefore, c(A) ‚ g C(A, σ, f ), σ .

Proof : The ¬rst equation is clear from the de¬nition of σ. The second equation

follows, since c x + σ(x) = f x + σ(x) .

Let c(A)0 = c(A) © Skew C(A, σ, f ), σ . As an intersection of Lie subalgebras,

c(A)0 is a subalgebra of L C(A, σ, f ) .

(8.17) Proposition. The map Id + σ : c(A) ’ F is a Lie-algebra homomorphism

which ¬ts into an exact sequence

Id+σ

0 ’ c(A)0 ’ c(A) ’ ’ F ’ 0.

’’

n(n’1)

In particular, it follows that dim c(A)0 = if deg A = n.

2

Proof : The de¬nition of c(A)0 shows that this set is the kernel of Id + σ. For x,

y ∈ A, we have c(x), c(y) = c x ’ σ(x), y by (??). The preceding lemma shows

that the image of this under Id + σ is equal to

TrdA x ’ σ(x), y = 0,

hence Id + σ is a Lie-algebra homomorphism.

Special features of the case where char F = 2 are collected in the following

proposition:

(8.18) Proposition. If char F = 2, there is a direct sum decomposition

c(A) = F • c(A)0 .

The restriction of δ to c(A)0 is an isomorphism of Lie algebras

∼

δ : c(A)0 ’ Alt(A, σ) = Skew(A, σ).

’

1 1

2 c, mapping x ∈ Skew(A, σ) to 2 c(x).

The inverse isomorphism is

Proof : The hypothesis that char F = 2 ensures that F © c(A)0 = {0}, hence c(A) =

F • c(A)0 . For a ∈ Alt(A, σ), we have

σ c(a) = c σ(a) = ’c(a)

by (??), hence c(a) ∈ c(A)0 . On the other hand, the de¬nition of δ yields

δ c(a) = a ’ σ(a) = 2a.

Since c(A)0 and Alt(A, σ) have the same dimension, it follows that δ is bijective

1

and that its inverse is 2 c.

(8.19) Example. Suppose A = Q1 — Q2 is a tensor product of two quaternion

algebras over a ¬eld F of arbitrary characteristic, and let σ = γ1 — γ2 be the tensor

product of the canonical involutions on Q1 and Q2 . Since γ1 and γ2 are symplec-

tic, there is a canonical quadratic pair (σ, f— ) on Q1 — Q2 : see (??). By (??) (if

char F = 2) or (??) (if char F = 2), the discriminant of (σ, f— ) is trivial, hence (??)

shows that C(A, σ, f— ) = C + — C ’ for some quaternion algebras C + , C ’ . More-

over, the canonical involution σ is symplectic (see (??)), hence it is the quaternion

98 II. INVARIANTS OF INVOLUTIONS

conjugation on C + and C ’ . We claim that C + and C ’ are isomorphic to Q1

and Q2 .

Let

(C + — C ’ ) = { (x+ , x’ ) ∈ C + — C ’ | TrdC + (x+ ) = TrdC ’ (x’ ) }

= { ξ ∈ C + — C ’ | TrdC + —C ’ (ξ) ∈ F }

and

(Q1 — Q2 ) = { (x1 , x2 ) ∈ Q1 — Q2 | TrdQ1 (x1 ) = TrdQ2 (x2 ) }.

In view of (??), we have c(A) ‚ (C + —C ’ ) , hence c(A) = (C + —C ’ ) by dimension

count. On the other hand, we may de¬ne a linear map ˜ : A ’ Q1 — Q2 by

˜(x1 — x2 ) = TrdQ2 (x2 )x1 , TrdQ1 (x1 )x2 for x1 ∈ Q1 , x2 ∈ Q2 .

Clearly, im ˜ ‚ (Q1 — Q2 ) ; the converse inclusion follows from the following ob-

servation: if (x1 , x2 ) ∈ Q1 — Q2 and TrdQ1 (x1 ) = TrdQ2 (x2 ) = ±, we have

˜(±’1 x1 — x2 ) if ± = 0,

(x1 , x2 ) =

˜(x1 — 2 + 1 — x2 ) if ± = 0,

where i ∈ Qi is an element of reduced trace 1 for i = 1, 2. A computation shows

that ˜ vanishes on the kernel of the canonical map c : A ’ C(A, σ, f— ) (see (??)),

hence it induces a surjective linear map c(A) ’ (Q1 — Q2 ) which we call again ˜.

Since c(A) and (Q1 — Q2 ) have the same dimension, this map is bijective:

∼

˜ : (C + — C ’ ) = c(A) ’ (Q1 — Q2 ) .

’

Using (??), one can check that this bijection is an isomorphism of Lie algebras. To

complete the proof, we show that this isomorphism extends to an isomorphism of

∼

(associative) F -algebras C + — C ’ = C(A, σ, f— ) ’ Q1 — Q2 . Since C + — C ’ is

’

generated by the subspace (C + — C ’ ) , the same argument as in the proof of (??)

shows that it su¬ces to ¬nd an isomorphism extending ˜ over an extension of F .

We may thus assume that Q1 and Q2 are split and identify Q1 = Q2 = EndF (V ) for

some 2-dimensional F -vector space V . Let b be a nonsingular alternating form on V

(such a form is uniquely determined up to a scalar factor) and let q be the quadratic

form on V — V whose polar bilinear form is b — b and such that q(v — w) = 0 for all

v, w ∈ V (see Exercise ?? of Chapter ??). The canonical quadratic pair (γ — γ, f— )

on A = EndF (V ) — EndF (V ) = EndF (V — V ) is then associated with the quadratic

form q, hence the standard identi¬cation •q induces an F -algebra isomorphism

∼

·q : C0 (V — V, q) ’ C(A, σ, f— )

’

(see (??)). By de¬nition of the canonical map c, we have

c(A) = ·q (V — V ) · (V — V ) .

On the other hand, the map i : V — V ’ M2 EndF (V ) de¬ned by

0 •b (v — w)

i(v — w) = for v, w ∈ V

’•b (w — v) 0

induces an F -algebra homomorphism i— : C(V — V, q) ’ M2 EndF (V ) by the

universal property of Cli¬ord algebras. This homomorphism is injective because

§8. THE CLIFFORD ALGEBRA 99

C(V — V, q) is simple, hence also surjective by dimension count. Under the isomor-

phism i— , the natural gradation of the Cli¬ord algebra corresponds to the checker-

board grading of M2 EndF (V ) , hence i— induces an F -algebra isomorphism

EndF (V ) 0

∼

i— : C0 (V — V, q) ’

’ EndF (V ) — EndF (V ).

0 EndF (V )

For v1 , v2 , w1 , w2 ∈ V , we have •q (v1 —w1 )—(v2 —w2 ) = •b (v1 —v2 )—•b (w1 —w2 ),

hence

˜ ·q (v1 — w1 · v2 — w2 ) =

= tr •b (w1 — w2 ) •b (v1 — v2 ), tr •b (v1 — v2 ) •b (w1 — w2 )

= b(w2 , w1 )•b (v1 — v2 ), b(v2 , v1 )•b (w1 — w2 ) .

On the other hand,

i— (v1 — w1 · v2 — w2 ) =

’•b (v1 — w1 ) —¦ •b (w2 — v2 ) 0

=

0 ’•b (w1 — v1 ) —¦ •b (v2 — w2 )

’b(w1 , w2 )•b (v1 — v2 ) 0

= .

0 ’b(v1 , v2 )•b (w1 — w2 )

Therefore, i— and ˜—¦·q have the same restriction to (V —V )·(V —V ), and it follows

∼

’1

that the F -algebra isomorphism i— —¦ ·q : C(A, σ, f— ) ’ EndF (V ) — EndF (V ) =

’

Q1 — Q2 extends ˜. This completes the proof of the claim.

In conclusion, we have shown:

C(Q1 — Q2 , γ1 — γ2 , f— ) Q1 — Q2 .

A more general statement is proved in (??) below.

8.D. The center of the Cli¬ord algebra. The center of the Cli¬ord algebra

C(A, σ, f ) of a central simple algebra A with a quadratic pair (σ, f ) is described

in (??) as an ´tale quadratic F -algebra. In this section, we show how elements of

e

the center can be produced explicitly, thus providing another proof of the second

part of (??).

We set Z(A, σ, f ) for the center of C(A, σ, f ). If char F = 2, the map f is

uniquely determined by σ and we use the shorter notation C(A, σ) for the Cli¬ord

algebra and Z(A, σ) for its center.

As may expected from (??), our methods in characteristic 2 and characteristic

not 2 are completely di¬erent. In characteristic di¬erent from 2 they rely on an

analogue of the pfa¬an, viewed as a map from Skew(A, σ) to Z(A, σ). The case

of characteristic 2 is simpler; it turns out then that Z(A, σ, f ) is in the image c(A)

of A in C(A, σ, f ) under the canonical map of §??.

Characteristic not 2. Our ¬rst result yields a standard form for certain skew-

symmetric elements in split algebras with orthogonal involution.

(8.20) Lemma. Let (V, q) be a nonsingular quadratic space of dimension n = 2m

over a ¬eld F of characteristic di¬erent from 2 and let a ∈ EndF (V ) satisfy σq (a) =

’a. Assume moreover that the characteristic polynomial of a splits into pairwise

distinct linear factors:

Pca (X) = (X ’ »1 )(X + »1 ) · · · (X ’ »m )(X + »m )

100 II. INVARIANTS OF INVOLUTIONS

for some »1 , . . . , »m ∈ F — . There exists an orthogonal basis (e1 , . . . , en ) of V such

that the matrix representing a with respect to this basis is

«

Λ1 0

0 »i

¬ ·

.. where Λi = »

. 0

i

0 Λm

Letting ±i = q(ei ) for i = 1, . . . , n, we have ±2i = ’±2i’1 for i = 1, . . . , m.

∼

Moreover, with •q : V — V ’ EndF (V ) the standard identi¬cation (??), we have

’

m

»i

a= •q (e2i’1 — e2i ’ e2i — e2i’1 ).

2±2i

i=1

Proof : For i = 1, . . . , m, let Vi ‚ V be the sum of the eigenspaces of a for

the eigenvalues »i and ’»i . The subspace Vi is thus the eigenspace of a2 for the

eigenvalue »2 . We have

i

V = V1 • · · · • V m

and the subspaces V1 , . . . , Vm are pairwise orthogonal since, for x ∈ Vi and y ∈ Vj ,

»2 bq (x, y) = bq a2 (x), y = bq x, a2 (y) = »2 bq (x, y),

i j

and »2 = »2 for i = j. It follows that the subspaces V1 , . . . , Vm are nonsingular.

i j

For i = 1, . . . , m, pick an anisotropic vector e2i’1 ∈ Vi and let e2i = »’1 a(e2i’1 ).

i

Since σq (a) = ’a, we have

bq (e2i’1 , e2i ) = »’1 bq e2i’1 , a(e2i’1 )

i

= ’»’1 bq a(e2i’1 ), e2i’1 = ’bq (e2i , e2i’1 ),

i

hence (e2i’1 , e2i ) is an orthogonal basis of Vi . It follows that (e1 , . . . , en ) is an

orthogonal basis of V , and the matrix of a with respect to this basis is as stated

above.

The equation a(e2i’1 ) = »i e2i yields

bq a(e2i’1 ), a(e2i’1 ) = »2 bq (e2i , e2i ) = 2»2 q(e2i ).

i i

On the other hand, since σq (a) = ’a and a2 (e2i’1 ) = »2 e2i’1 , the left side is also

i

equal to

bq e2i’1 , ’a2 (e2i’1 ) = ’»2 bq (e2i’1 , e2i’1 ) = ’2»2 q(e2i’1 ),

i i

hence q(e2i ) = ’q(e2i’1 ). Finally, for i, j = 1, . . . , m we have

’2q(e2i’1 )e2i if i = j,

•q (e2i’1 — e2i ’ e2i — e2i’1 )(e2j’1 ) =

0 if i = j,

and

2q(e2i )e2i’1 if i = j,

•q (e2i’1 — e2i ’ e2i — e2i’1 )(e2j ) =

0 if i = j.

The last equation in the statement of the lemma follows.

Let (A, σ) be a central simple algebra with orthogonal involution over a ¬eld F

of characteristic di¬erent from 2. We assume throughout this subsection that the

degree of A is even and let deg A = n = 2m. Our ¬rst observations also require

the ¬eld F to be in¬nite. Under this hypothesis, we denote by S(A, σ) the set of

§8. THE CLIFFORD ALGEBRA 101

skew-symmetric units in A— whose reduced characteristic polynomials are separable

(i.e., have no repeated root in an algebraic closure). This set is Zariski-open in

Skew(A, σ), since it is de¬ned by the condition that the discriminant of the reduced

characteristic polynomial does not vanish. By scalar extension to a splitting ¬eld L

such that (AL , σL ) Mn (L), t , we can see that this open set is not empty, since

S Mn (L), t = ….

Over an algebraic closure, the reduced characteristic polynomial of every a ∈

S(A, σ) splits into a product of pairwise distinct linear factors of the form

Prda (X) = (X ’ »1 )(X + »1 ) · · · (X ’ »m )(X + »m )

since σ(a) = ’a. Therefore, the subalgebra F [a] ‚ A generated by a has dimen-

sion n and F [a2 ] = F [a]©Sym(A, σ) has dimension m. Clearly, F [a]©Skew(A, σ) =

a · F [a2 ].

(8.21) Lemma. Let a ∈ S(A, σ). Denote H = F [a] © Skew(A, σ) = a · F [a2 ] and

E = F [a2 ]. The bilinear form T : H — H ’ F de¬ned by

T (x, y) = TE/F (xy) for x, y ∈ H

is nonsingular. Moreover, the elements in the image c(H) of H in C(A, σ) under

the canonical map c : A ’ C(A, σ) commute.

Proof : It su¬ces to check the lemma over a scalar extension. We may therefore

assume that A and the reduced characteristic polynomial of a are split12 . We

identify (A, σ) = EndF (V ), σq for some nonsingular quadratic space (V, q) of

dimension n. By (??), there is an orthogonal basis (e1 , . . . , en ) of V such that,

letting q(ei ) = ±i ∈ F — ,

m

»i

a= •q (e2i’1 — e2i ’ e2i — e2i’1 )

2±2i

i=1

(and ±2i’1 = ’±2i for i = 1, . . . , m). For i = 1, . . . , m, let

1

hi = •q (e2i’1 — e2i ’ e2i — e2i’1 ) ∈ A.

2±2i

By using the matrix representation with respect to the basis (e1 , . . . , en ), it is easily

seen that every skew-symmetric element in A which commutes with a is a linear

combination of h1 , . . . , hm . In particular, H is contained in the span of h1 , . . . , hm .

Since dim H = dim F [a2 ] = m, it follows that (h1 , . . . , hm ) is a basis of H.

With the same matrix representation, it is easy to check that h2 , . . . , h2 are

1 m

primitive orthogonal idempotents in E which form a basis of E and that hi hj = 0

for i = j. Therefore, the bilinear form T satis¬es

m m m

T( xi h i , y j hj ) = xi y i

i=1 j=1 i=1

for x1 , . . . , xm , y1 , . . . , ym ∈ F . It is therefore nonsingular.

Consider now the subspace c(H) of C(A, σ) spanned by c(h1 ), . . . , c(hm ). For

i = 1, . . . , m we have

1 1

c(hi ) = (e2i’1 · e2i ’ e2i · e2i’1 ) = e2i’1 · e2i .

2±2i ±2i

These elements commute, since (e1 , . . . , en ) is an orthogonal basis of V .

12 In fact, the algebra A splits as soon as the reduced characteristic polynomial of a splits.

102 II. INVARIANTS OF INVOLUTIONS

Since the bilinear form T on H is nonsingular, every linear form on H is of the

type x ’ TE/F (hx) for some h ∈ H. Therefore, every homogeneous polynomial

map P : H ’ F of degree d has the form

P (x) = TE/F (hi1 x) · · · TE/F (hid x)

i

for some hi1 , . . . , hid ∈ H, and in the d-th symmetric power S d H the element

i hi1 · · · hid is uniquely determined by P . In particular, there is a uniquely deter-

mined element ν = i hi1 · · · him ∈ S m H such that

NE/F (ax) = TE/F (hi1 x) · · · TE/F (him x) for x ∈ H,

i

since the map x ’ NE/F (ax) is a homogeneous polynomial map of degree m on H.

Since the elements in c(H) commute, the canonical map c induces a well-de¬ned

linear map S m H ’ C(A, σ). We set π(a) for the image under this induced map of

the element ν ∈ S m H de¬ned above.

In summary, the element π(a) ∈ C(A, σ) is de¬ned as follows:

(8.22) De¬nition. For a ∈ S(A, σ), we let

π(a) = c(hi1 ) · · · c(him )

i

where hi1 , . . . , him ∈ H = a · F [a2 ] satisfy

NF [a2 ]/F (ax) = TF [a2 ]/F (hi1 x) · · · TF [a2 ]/F (him x)

i

for all x ∈ a · F [a2 ].

(8.23) Lemma. Let ι be the nontrivial automorphism of the center Z(A, σ) of

C(A, σ). For a ∈ S(A, σ) we have π(a) ∈ Z(A, σ), ι π(a) = ’π(a) and

π(a)2 = (’1)m NrdA (a).

Proof : It su¬ces to verify the assertions over a scalar extension. We may thus

assume that A and the reduced characteristic polynomial of a are split, and use the

same notation as in (??). In particular, we let (A, σ) = EndF (V ), σq and choose

an orthogonal basis (e1 , . . . , em ) of V such that

1

m

a= »i hi where hi = •q (e2i’1 — e2i ’ e2i — e2i’1 ).

i=1

2±2i

m

As observed in (??), the elements h1 , . . . , hm form a basis of H. For x = xi h i ,

i=1

we have

m

» i xi h 2 ) = » 1 · · · » m x1 · · · x m .

NF [a2 ]/F (ax) = NF [a2 ]/F ( i

i=1

On the other hand, TF [a2 ]/F (hi x) = xi , hence ν = h1 . . . hm and

π(a) = »1 · · · »m c(h1 ) · · · c(hm ).

It was also seen in (??) that c(hi ) = ±1 e2i’1 · e2i where ±2i = q(e2i ) = ’q(e2i’1 ),

2i

hence

»1 · · · » m

π(a) = m e1 · · · en .

i=1 ±2i

It is then clear that π(a) ∈ Z(A, σ) and ι π(a) = ’π(a).

m

Since (e1 · · · en )2 = (’1)m e2 · · · e2 = i=1 ±2 and NrdA (a) = (’1)m »2 · · · »2 ,

n m

1 1

2i

the last equation in the statement of the proposition follows.

§8. THE CLIFFORD ALGEBRA 103

To extend the de¬nition of π to the whole of Skew(A, σ), including also the

case where the base ¬eld F is ¬nite (of characteristic di¬erent from 2), we adjoin

indeterminates to F and apply π to a generic skew-symmetric element.

Pick a basis (a1 , . . . , ad ) of Skew(A, σ) where d = m(n ’ 1) = dim Skew(A, σ),

d

and let ξ = i=1 ai xi where x1 , . . . , xd are indeterminates over F . We have

ξ ∈ S AF (x1 ,...,xd ) , σF (x1 ,...,xd ) and

π(ξ) ∈ Z(AF (x1 ,...,xd ) , σF (x1 ,...,xd ) ) = Z(A, σ) —F F (x1 , . . . , xd ).

Since π(ξ)2 = Nrd(ξ) is a polynomial in x1 , . . . , xd , we have in fact

π(ξ) ∈ Z(A, σ) — F [x1 , . . . , xd ].

We may then de¬ne π(a) for all a ∈ Skew(A, σ) by specializing the indeterminates.

We call the map π : Skew(A, σ) ’ Z(A, σ) thus de¬ned the generalized pfa¬an of

(A, σ) in view of Example (??) below.

(8.24) Proposition. The map π : Skew(A, σ) ’ Z(A, σ) is a homogeneous poly-

nomial map of degree m. Denoting by ι the nontrivial automorphism of Z(A, σ)

over F , we have

π(a)2 = (’1)m NrdA (a)

ι π(a) = ’π(a) and

for all a ∈ Skew(A, σ). Moreover, for all x ∈ A, a ∈ Skew(A, σ),

π xaσ(x) = NrdA (x)π(a).

Proof : For the generic element ξ we have by (??)

and π(ξ)2 = (’1)m Nrd(ξ).

ι π(ξ) = ’π(ξ)

The same formulas follow for all a ∈ Skew(A, σ) by specialization. Since the reduced

norm is a homogeneous polynomial map of degree n, the second formula shows that

π is a homogeneous polynomial map of degree m. It also shows that an element

a ∈ Skew(A, σ) is invertible if and only if π(a) = 0.

In order to prove the last property, ¬x some element a ∈ Skew(A, σ). If a is

not invertible, then π(a) = π xaσ(x) = 0 for all x ∈ A and the property is clear.

Suppose a ∈ A— . Since the F -vector space of elements z ∈ Z(A, σ) such that

ι(z) = ’z has dimension 1, we have for all x ∈ A

π xaσ(x) = P (x)π(a) for some P (x) ∈ F .

The map P : A ’ F is polynomial and satis¬es

2

π xaσ(x) Nrd xaσ(x)

2

= Nrd(x)2 .

P (x) = =

π(a)2 Nrd(a)

By adjoining indeterminates to F if necessary, we may assume F is in¬nite. The

algebra of polynomial maps on A is then a domain, hence the preceding equation

yields the alternative: P (x) = Nrd(x) for all x or P (x) = ’ Nrd(x) for all x. Since

P (1) = 1, we have P (x) = Nrd(x) for all x ∈ A.

Using the map π, we may give an alternate proof of (??) and of part of (??):

(8.25) Corollary. For all a, b ∈ Skew(A, σ) © A— ,

NrdA (a) ≡ NrdA (b) mod F —2

F [X]/ X 2 ’ (’1)m NrdA (a) .

and Z(A, σ)

104 II. INVARIANTS OF INVOLUTIONS

Proof : Since the F -vector space of elements z ∈ Z(A, σ) such that ι(z) = ’z is 1-

dimensional, we have π(a) ≡ π(b) mod F — . By squaring both sides we obtain the

¬rst equation. The second equation follows from the fact that Z(A, σ) = F π(a) .

(8.26) Example. In the case where A is split, the map π can be described ex-

plicitly in terms of the pfa¬an. Let (V, q) be a nonsingular quadratic space of

dimension n = 2m over F . For (A, σ) = EndF (V ), σq , we identify A = V — V as

in (??) and C(A, σ) = C0 (V, q) as in (??). Let (e1 , . . . , en ) be an orthogonal basis

1

of V . The elements 2 (ei — ej ’ ej — ei ) for 1 ¤ i < j ¤ n form a basis of Skew(A, σ).

aij

For a = i<j 2 (ei — ej ’ ej — ei ) ∈ Skew(A, σ), de¬ne a skew-symmetric matrix

a = (aij ) ∈ Mn (F ) by

±

aij if i < j,

aij = 0 if i = j

’aji if i > j.

Claim. The element π(a) ∈ Z(A, σ) is related to the pfa¬an pf(a ) as follows:

π(a) = pf(a )e1 · · · en .

Proof : A computation shows that the matrix representing a with respect to the

basis (e1 , . . . , en ) is a · d where

«

q(e1 ) 0

¬ ·

..

d=

.

0 q(en )

Since det a = pf(a )2 , it follows that

2

det a = pf(a )2 q(e1 ) · · · q(en ) = (’1)m pf(a )e1 · · · en .

Therefore, π(a) = ± pf(a )e1 · · · en , since both sides are polynomial maps of de-

gree m whose squares are equal. To prove that the equality holds with the + sign,

it su¬ces to evaluate both sides on a particular unit in Skew(A, σ). Adjoin inde-

terminates z1 , . . . , zm to F and consider

m

zi

ζ= (e2i’1 — e2i ’ e2i — e2i’1 ) ∈ Skew AF (z1 ,...,zm ) , σF (z1 ,...,zm ) .

2

i=1

The same computation as in (??) shows that

π(ζ) = z1 · · · zm e1 · · · en .

m1

By setting z1 = · · · = zm = 1, we get for a = i=1 2 (e2i’1 — e2i ’ e2i — e2i’1 ):

π(a) = e1 · · · en .

On the other hand, the corresponding matrix a is

«

J 0

01

¬ ·

..

a = where J = ,

. ’1 0

0 J

hence pf(a ) = 1.

§8. THE CLIFFORD ALGEBRA 105

Characteristic 2. Let (σ, f ) be a quadratic pair on a central simple algebra A

over a ¬eld F of characteristic 2. We write Z(A, σ, f ) for the center of the Cli¬ord

algebra C(A, σ, f ) and by ι the nontrivial automorphism of Z(A, σ, f ) over F .

Consider the set

Λ = { ∈ A | f (s) = TrdA ( s) for s ∈ Sym(A, σ) }.

By (??), this set is nonempty; it is a coset of Alt(A, σ).

The following proposition shows that the canonical map c : A ’ C(A, σ, f ),

restricted to Λ, plays a rˆle analogous to the map π in characteristic di¬erent

o

from 2.